12,080
edits
Changes
add image stub
Figure 12. The Anchor
Figure 12. The Anchor
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]</p>
<p><center><font size="+1">'''Figure 12. The Anchor'''</font></center></p>
===Figure 13. The Tiller===
===Figure 13. The Tiller===
Figure 13. The Tiller
Figure 13. The Tiller
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]</p>
<p><center><font size="+1">'''Figure 13. The Tiller'''</font></center></p>
===Table 14. Differential Propositions===
===Table 14. Differential Propositions===
|}
|}
</font><br>
</font><br>
===Figure 16. A Couple of Fourth Gear Orbits===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]</p>
<p><center><font size="+1">'''Figure 16. A Couple of Fourth Gear Orbits'''</font></center></p>
===Figure 16-a. A Couple of Fourth Gear Orbits: 1===
===Figure 16-a. A Couple of Fourth Gear Orbits: 1===
Figure 18-a. Extension from 1 to 2 Dimensions: Areal
Figure 18-a. Extension from 1 to 2 Dimensions: Areal
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 18-a. Extension from 1 to 2 Dimensions: Areal'''</font></center></p>
===Figure 18-b. Extension from 1 to 2 Dimensions: Bundle===
===Figure 18-b. Extension from 1 to 2 Dimensions: Bundle===
Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 18-b. Extension from 1 to 2 Dimensions: Bundle'''</font></center></p>
===Figure 18-c. Extension from 1 to 2 Dimensions: Compact===
===Figure 18-c. Extension from 1 to 2 Dimensions: Compact===
Figure 18-c. Extension from 1 to 2 Dimensions: Compact
Figure 18-c. Extension from 1 to 2 Dimensions: Compact
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 18-c. Extension from 1 to 2 Dimensions: Compact'''</font></center></p>
===Figure 18-d. Extension from 1 to 2 Dimensions: Digraph===
===Figure 18-d. Extension from 1 to 2 Dimensions: Digraph===
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 18-d. Extension from 1 to 2 Dimensions: Digraph'''</font></center></p>
===Figure 19-a. Extension from 2 to 4 Dimensions: Areal===
===Figure 19-a. Extension from 2 to 4 Dimensions: Areal===
Figure 19-a. Extension from 2 to 4 Dimensions: Areal
Figure 19-a. Extension from 2 to 4 Dimensions: Areal
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 19-a. Extension from 2 to 4 Dimensions: Areal'''</font></center></p>
===Figure 19-b. Extension from 2 to 4 Dimensions: Bundle===
===Figure 19-b. Extension from 2 to 4 Dimensions: Bundle===
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 19-b. Extension from 2 to 4 Dimensions: Bundle'''</font></center></p>
===Figure 19-c. Extension from 2 to 4 Dimensions: Compact===
===Figure 19-c. Extension from 2 to 4 Dimensions: Compact===
Figure 19-c. Extension from 2 to 4 Dimensions: Compact
Figure 19-c. Extension from 2 to 4 Dimensions: Compact
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 19-c. Extension from 2 to 4 Dimensions: Compact'''</font></center></p>
===Figure 19-d. Extension from 2 to 4 Dimensions: Digraph===
===Figure 19-d. Extension from 2 to 4 Dimensions: Digraph===
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 19-d. Extension from 2 to 4 Dimensions: Digraph'''</font></center></p>
===Figure 20-i. Thematization of Conjunction (Stage 1)===
===Figure 20-i. Thematization of Conjunction (Stage 1)===
Figure 20-i. Thematization of Conjunction (Stage 1)
Figure 20-i. Thematization of Conjunction (Stage 1)
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]</p>
<p><center><font size="+1">'''Figure 20-i. Thematization of Conjunction (Stage 1)'''</font></center></p>
===Figure 20-ii. Thematization of Conjunction (Stage 2)===
===Figure 20-ii. Thematization of Conjunction (Stage 2)===
Figure 20-ii. Thematization of Conjunction (Stage 2)
Figure 20-ii. Thematization of Conjunction (Stage 2)
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]</p>
<p><center><font size="+1">'''Figure 20-ii. Thematization of Conjunction (Stage 2)'''</font></center></p>
===Figure 20-iii. Thematization of Conjunction (Stage 3)===
===Figure 20-iii. Thematization of Conjunction (Stage 3)===
Figure 20-iii. Thematization of Conjunction (Stage 3)
Figure 20-iii. Thematization of Conjunction (Stage 3)
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]</p>
<p><center><font size="+1">'''Figure 20-iii. Thematization of Conjunction (Stage 3)'''</font></center></p>
===Figure 21. Thematization of Disjunction and Equality===
===Figure 21. Thematization of Disjunction and Equality===
Figure 21. Thematization of Disjunction and Equality
Figure 21. Thematization of Disjunction and Equality
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]</p>
<p><center><font size="+1">'''Figure 21. Thematization of Disjunction and Equality'''</font></center></p>
===Table 22. Disjunction ''f'' and Equality ''g''===
===Table 22. Disjunction ''f'' and Equality ''g''===
Figure 30. Generic Frame of a Logical Transformation
Figure 30. Generic Frame of a Logical Transformation
</pre>
</pre>
'''Note.''' The following image was corrupted in transit between software platforms.
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 30 -- Generic Frame of a Logical Transformation.gif|center]]</p>
<p><center><font size="+1">'''Figure 30. Generic Frame of a Logical Transformation'''</font></center></p>
===Formula Display 3===
===Formula Display 3===
Figure 31. Operator Diagram (1)
Figure 31. Operator Diagram (1)
</pre>
</pre>
'''Note.''' The following image was corrupted in transit between software platforms.
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 31 -- Operator Diagram (1).gif|center]]</p>
<p><center><font size="+1">'''Figure 31. Operator Diagram (1)'''</font></center></p>
===Figure 32. Operator Diagram (2)===
===Figure 32. Operator Diagram (2)===
Figure 32. Operator Diagram (2)
Figure 32. Operator Diagram (2)
</pre>
</pre>
'''Note.''' The following image was corrupted in transit between software platforms.
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 32 -- Operator Diagram (2).gif|center]]</p>
<p><center><font size="+1">'''Figure 32. Operator Diagram (2)'''</font></center></p>
===Figure 33-i. Analytic Diagram (1)===
===Figure 33-i. Analytic Diagram (1)===
Figure 33-i. Analytic Diagram (1)
Figure 33-i. Analytic Diagram (1)
</pre>
</pre>
'''Note.''' The following image was corrupted in transit between software platforms.
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 33-i -- Analytic Diagram (1).gif|center]]</p>
<p><center><font size="+1">'''Figure 33-i. Analytic Diagram (1)'''</font></center></p>
===Figure 33-ii. Analytic Diagram (2)===
===Figure 33-ii. Analytic Diagram (2)===
Figure 33-ii. Analytic Diagram (2)
Figure 33-ii. Analytic Diagram (2)
</pre>
</pre>
'''Note.''' The following image was corrupted in transit between software platforms.
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 33-ii -- Analytic Diagram (2).gif|center]]</p>
<p><center><font size="+1">'''Figure 33-ii. Analytic Diagram (2)'''</font></center></p>
===Formula Display 4===
===Formula Display 4===
Figure 34. Tangent Functor Diagram
Figure 34. Tangent Functor Diagram
</pre>
</pre>
'''Note.''' The following image was corrupted in transit between software platforms.
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 34 -- Tangent Functor Diagram.gif|center]]</p>
<p><center><font size="+1">'''Figure 34. Tangent Functor Diagram'''</font></center></p>
===Figure 35. Conjunction as Transformation===
===Figure 35. Conjunction as Transformation===
Figure 35. Conjunction as Transformation
Figure 35. Conjunction as Transformation
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]</p>
<p><center><font size="+1">'''Figure 35. Conjunction as Transformation'''</font></center></p>
===Table 36. Computation of !e!J===
===Table 36. Computation of !e!J===
</font><br>
</font><br>
===Figure 37-a. Tacit Extension of J (Areal)===
===Figure 37-a. Tacit Extension of ''J'' (Areal)===
<pre>
<pre>
</pre>
</pre>
===Figure 37-b. Tacit Extension of J (Bundle)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 37-a. Tacit Extension of ''J'' (Areal)'''</font></center></p>
===Figure 37-b. Tacit Extension of ''J'' (Bundle)===
<pre>
<pre>
</pre>
</pre>
===Figure 37-c. Tacit Extension of J (Compact)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 37-b. Tacit Extension of ''J'' (Bundle)'''</font></center></p>
===Figure 37-c. Tacit Extension of ''J'' (Compact)===
<pre>
<pre>
</pre>
</pre>
===Figure 37-d. Tacit Extension of J (Digraph)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 37-c. Tacit Extension of ''J'' (Compact)'''</font></center></p>
===Figure 37-d. Tacit Extension of ''J'' (Digraph)===
<pre>
<pre>
Figure 37-d. Tacit Extension of J (Digraph)
Figure 37-d. Tacit Extension of J (Digraph)
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 37-d. Tacit Extension of ''J'' (Digraph)'''</font></center></p>
===Table 38. Computation of EJ (Method 1)===
===Table 38. Computation of EJ (Method 1)===
</font><br>
</font><br>
===Figure 40-a. Enlargement of J (Areal)===
===Figure 40-a. Enlargement of ''J'' (Areal)===
<pre>
<pre>
</pre>
</pre>
===Figure 40-b. Enlargement of J (Bundle)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 40-a. Enlargement of ''J'' (Areal)'''</font></center></p>
===Figure 40-b. Enlargement of ''J'' (Bundle)===
<pre>
<pre>
</pre>
</pre>
===Figure 40-c. Enlargement of J (Compact)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 40-b. Enlargement of ''J'' (Bundle)'''</font></center></p>
===Figure 40-c. Enlargement of ''J'' (Compact)===
<pre>
<pre>
</pre>
</pre>
===Figure 40-d. Enlargement of J (Digraph)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 40-c. Enlargement of ''J'' (Compact)'''</font></center></p>
===Figure 40-d. Enlargement of ''J'' (Digraph)===
<pre>
<pre>
Figure 40-d. Enlargement of J (Digraph)
Figure 40-d. Enlargement of J (Digraph)
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 40-d. Enlargement of ''J'' (Digraph)'''</font></center></p>
===Table 41. Computation of DJ (Method 1)===
===Table 41. Computation of DJ (Method 1)===
</font><br>
</font><br>
===Figure 44-a. Difference Map of J (Areal)===
===Figure 44-a. Difference Map of ''J'' (Areal)===
<pre>
<pre>
</pre>
</pre>
===Figure 44-b. Difference Map of J (Bundle)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 44-a. Difference Map of ''J'' (Areal)'''</font></center></p>
===Figure 44-b. Difference Map of ''J'' (Bundle)===
<pre>
<pre>
</pre>
</pre>
===Figure 44-c. Difference Map of J (Compact)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 44-b. Difference Map of ''J'' (Bundle)'''</font></center></p>
===Figure 44-c. Difference Map of ''J'' (Compact)===
<pre>
<pre>
</pre>
</pre>
===Figure 44-d. Difference Map of J (Digraph)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 44-c. Difference Map of ''J'' (Compact)'''</font></center></p>
===Figure 44-d. Difference Map of ''J'' (Digraph)===
<pre>
<pre>
Figure 44-d. Difference Map of J (Digraph)
Figure 44-d. Difference Map of J (Digraph)
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 44-d. Difference Map of ''J'' (Digraph)'''</font></center></p>
===Table 45. Computation of dJ===
===Table 45. Computation of dJ===
</font><br>
</font><br>
===Figure 46-a. Differential of J (Areal)===
===Figure 46-a. Differential of ''J'' (Areal)===
<pre>
<pre>
</pre>
</pre>
===Figure 46-b. Differential of J (Bundle)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 46-a. Differential of ''J'' (Areal)'''</font></center></p>
===Figure 46-b. Differential of ''J'' (Bundle)===
<pre>
<pre>
</pre>
</pre>
===Figure 46-c. Differential of J (Compact)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 46-b. Differential of ''J'' (Bundle)'''</font></center></p>
===Figure 46-c. Differential of ''J'' (Compact)===
<pre>
<pre>
</pre>
</pre>
===Figure 46-d. Differential of J (Digraph)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 46-c. Differential of ''J'' (Compact)'''</font></center></p>
===Figure 46-d. Differential of ''J'' (Digraph)===
<pre>
<pre>
Figure 46-d. Differential of J (Digraph)
Figure 46-d. Differential of J (Digraph)
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 46-d. Differential of ''J'' (Digraph)'''</font></center></p>
===Table 47. Computation of rJ===
===Table 47. Computation of rJ===
</font><br>
</font><br>
===Figure 48-a. Remainder of J (Areal)===
===Figure 48-a. Remainder of ''J'' (Areal)===
<pre>
<pre>
</pre>
</pre>
===Figure 48-b. Remainder of J (Bundle)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 48-a. Remainder of ''J'' (Areal)'''</font></center></p>
===Figure 48-b. Remainder of ''J'' (Bundle)===
<pre>
<pre>
</pre>
</pre>
===Figure 48-c. Remainder of J (Compact)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 48-b. Remainder of ''J'' (Bundle)'''</font></center></p>
===Figure 48-c. Remainder of ''J'' (Compact)===
<pre>
<pre>
</pre>
</pre>
===Figure 48-d. Remainder of J (Digraph)===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 48-c. Remainder of ''J'' (Compact)'''</font></center></p>
===Figure 48-d. Remainder of ''J'' (Digraph)===
<pre>
<pre>
Figure 48-d. Remainder of J (Digraph)
Figure 48-d. Remainder of J (Digraph)
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 48-d. Remainder of ''J'' (Digraph)'''</font></center></p>
===Table 49. Computation Summary for J===
===Table 49. Computation Summary for J===
Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)
Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]</p>
<p><center><font size="+1">'''Figure 52. Decomposition of E''J'''''</font></center></p>
===Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)===
===Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)===
Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]</p>
<p><center><font size="+1">'''Figure 53. Decomposition of D''J'''''</font></center></p>
===Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators===
===Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators===
| Transformation, or Mapping
| Transformation, or Mapping
| ['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
| ['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
|-
|-
| valign="top" |
| valign="top" |
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <math>\epsilon</math> :
|-
|-
|
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
|
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → ''X''<sup> •</sup>)
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <math>\epsilon</math>''J'' :
|-
|-
|
| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''
|-
|-
|
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''B'''
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <math>\epsilon</math>''J'' :
|-
|-
|
| [''u'', ''v'', d''u'', d''v''] → [''x'']
|-
|-
|
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
|}
|}
|-
|-
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <math>\eta</math> :
|-
|-
|
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
|
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <math>\eta</math>''J'' :
|-
|-
|
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
|
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <math>\eta</math>''J'' :
|-
|-
|
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
|-
|
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|}
|-
|-
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| E :
|-
|-
|
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
|
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| E''J'' :
|-
|-
|
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
|
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| E''J'' :
|-
|-
|
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
|-
|
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|}
|-
|-
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| D :
|-
|-
|
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
|
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| D''J'' :
|-
|-
|
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
|
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| D''J'' :
|-
|-
|
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
|-
|
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|}
|-
|-
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| d :
|-
|-
|
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
|
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| d''J'' :
|-
|-
|
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
|
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| d''J'' :
|-
|-
|
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
|-
|
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|}
|-
|-
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| r :
|-
|-
|
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
|
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| r''J'' :
|-
|-
|
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
|
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| r''J'' :
|-
|-
|
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
|-
|
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|}
|-
|-
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
|-
|-
|
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
|
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|}
|
|
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <font face=georgia>'''e'''</font>''J'' :
|-
|-
|
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
|-
|
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|}
|-
|-
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
|-
|-
|
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
|
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|}
|
|
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <font face=georgia>'''E'''</font>''J'' :
|-
|-
|
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
|-
|
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|}
|-
|-
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
|-
|-
|
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
|
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|}
|
|
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <font face=georgia>'''D'''</font>''J'' :
|-
|-
|
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
|-
|
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|}
|-
|-
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
|-
|-
|
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
|-
|
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| d''J'' :
|-
|-
|
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
|-
|
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|}
|
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <font face=georgia>'''T'''</font>''J'' :
|-
|-
|
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
|-
|
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|}
|}
|}<br>
<pre>
===Figure 56-a1. Radius Map of the Conjunction J = uv===
===Figure 56-a1. Radius Map of the Conjunction J = uv===
Figure 56-a1. Radius Map of the Conjunction J = uv
Figure 56-a1. Radius Map of the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-a1. Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-a2. Secant Map of the Conjunction J = uv===
===Figure 56-a2. Secant Map of the Conjunction J = uv===
Figure 56-a2. Secant Map of the Conjunction J = uv
Figure 56-a2. Secant Map of the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-a2. Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-a3. Chord Map of the Conjunction J = uv===
===Figure 56-a3. Chord Map of the Conjunction J = uv===
Figure 56-a3. Chord Map of the Conjunction J = uv
Figure 56-a3. Chord Map of the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-a3. Chord Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-a4. Tangent Map of the Conjunction J = uv===
===Figure 56-a4. Tangent Map of the Conjunction J = uv===
Figure 56-a4. Tangent Map of the Conjunction J = uv
Figure 56-a4. Tangent Map of the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-a4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-b1. Radius Map of the Conjunction J = uv===
===Figure 56-b1. Radius Map of the Conjunction J = uv===
Figure 56-b1. Radius Map of the Conjunction J = uv
Figure 56-b1. Radius Map of the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-b1. Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-b2. Secant Map of the Conjunction J = uv===
===Figure 56-b2. Secant Map of the Conjunction J = uv===
Figure 56-b2. Secant Map of the Conjunction J = uv
Figure 56-b2. Secant Map of the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-b2. Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-b3. Chord Map of the Conjunction J = uv===
===Figure 56-b3. Chord Map of the Conjunction J = uv===
Figure 56-b3. Chord Map of the Conjunction J = uv
Figure 56-b3. Chord Map of the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-b3. Chord Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 56-b4. Tangent Map of the Conjunction J = uv===
===Figure 56-b4. Tangent Map of the Conjunction J = uv===
Figure 56-b4. Tangent Map of the Conjunction J = uv
Figure 56-b4. Tangent Map of the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-b4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 57-1. Radius Operator Diagram for the Conjunction J = uv===
===Figure 57-1. Radius Operator Diagram for the Conjunction J = uv===
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]</p>
<p><center><font size="+1">'''Figure 57-1. Radius Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 57-2. Secant Operator Diagram for the Conjunction J = uv===
===Figure 57-2. Secant Operator Diagram for the Conjunction J = uv===
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]</p>
<p><center><font size="+1">'''Figure 57-2. Secant Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 57-3. Chord Operator Diagram for the Conjunction J = uv===
===Figure 57-3. Chord Operator Diagram for the Conjunction J = uv===
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]</p>
<p><center><font size="+1">'''Figure 57-3. Chord Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
===Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv===
===Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv===
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
</pre>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]</p>
<p><center><font size="+1">'''Figure 57-4. Tangent Functor Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
===Formula Display 11===
===Formula Display 11===
</pre>
</pre>
===Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes===
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators'''
|- style="background:paleturquoise"
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
! Item
! Notation
| | Operator | Proposition | Transformation |
! Description
| | or | or | or |
! Type
| | Operand | Component | Mapping |
|-
| valign="top" | ''U''<sup> •</sup>
| valign="top" | <font face="courier new">= </font>[''u'', ''v'']
| Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
| valign="top" | Source Universe
| | | | |
| valign="top" | ['''B'''<sup>''n''</sup>]
| | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k |
|-
| | | | |
| valign="top" | ''X''<sup> •</sup>
| valign="top" |
| | | | |
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face="courier new">= </font>[''x'', ''y'']
|-
| | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] |
| <font face="courier new">= </font>[''f'', ''g'']
| | | | |
|}
| valign="top" | Target Universe
| | | | |
| valign="top" | ['''B'''<sup>''k''</sup>]
| Trope | !h! : | !h!F_i : | !h!F : |
|-
| valign="top" | E''U''<sup> •</sup>
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| valign="top" | <font face="courier new">= </font>[''u'', ''v'', d''u'', d''v'']
| | | | |
| valign="top" | Extended Source Universe
| valign="top" | ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>]
| | | | |
|-
| Enlargement | E : | EF_i : | EF : |
| valign="top" | E''X''<sup> •</sup>
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| valign="top" |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| | | | |
| <font face="courier new">= </font>[''x'', ''y'', d''x'', d''y'']
|-
| | | | |
| <font face="courier new">= </font>[''f'', ''g'', d''f'', d''g'']
|}
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| valign="top" | Extended Target Universe
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| valign="top" | ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
| | | | |
|-
| ''F''
| | | | |
| ''F'' = ‹''f'', ''g''› : ''U''<sup> •</sup> → ''X''<sup> •</sup>
| Differential | d : | dF_i : | dF : |
| Transformation, or Mapping
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]
|-
| | | | |
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| | | | |
|
| Remainder | r : | rF_i : | rF : |
|-
| ''f''
|-
| ''g''
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| ''f'', ''g'' : ''U'' → '''B'''
|-
| ''f'' : ''U'' → [''x''] ⊆ ''X''<sup> •</sup>
|-
| ''g'' : ''U'' → [''y''] ⊆ ''X''<sup> •</sup>
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Proposition
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
| '''B'''<sup>''n''</sup> → '''B'''
|-
| ∈ ('''B'''<sup>''n''</sup>, '''B'''<sup>''n''</sup> → '''B''')
|-
| = ('''B'''<sup>''n''</sup> +→ '''B''') = ['''B'''<sup>''n''</sup>]
|}
|-
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| W
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| W :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> ,
|-
| ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
|-
| →
|-
| (E''U''<sup> •</sup> → E''X''<sup> •</sup>) ,
|-
| for each W in the set:
|-
| {<math>\epsilon</math>, <math>\eta</math>, E, D, d}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Operator
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
|
|-
| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
|-
| ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
|-
| (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
|-
| →
|-
| (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>])
|-
|
|
|-
|-
|
|
|}
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\epsilon</math>
|-
| <math>\eta</math>
|-
| E
|-
| D
|-
| d
|}
|}
| valign="top" |
| colspan="2" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
| Tacit Extension Operator || <math>\epsilon</math>
|-
| Trope Extension Operator || <math>\eta</math>
|-
| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |
| Enlargement Operator || E
| |
|-
| Difference Operator || D
|-
| Differential Operator || d
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
| ‹''x'', ''y''›
| =
| ''F''‹''u'', ''v''›
| =
| ‹((''u'')(''v'')), ((''u'', ''v''))›
|}
|}
|-
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''W'''</font>
|}
|}
</font><br>
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
<br><font face="courier new">
| <font face=georgia>'''W'''</font> :
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|-
|
| ''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''U''<sup> •</sup> = E''U''<sup> •</sup> ,
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|-
|
| ''X''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup> = E''X''<sup> •</sup> ,
| ‹''x'', ''y''›
|-
| =
| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
| ''F''‹''u'', ''v''›
|-
| =
| →
| ‹((''u'')(''v'')), ((''u'', ''v''))›
|-
| (<font face=georgia>'''T'''</font>''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup>) ,
|-
| for each <font face=georgia>'''W'''</font> in the set:
|-
| {<font face=georgia>'''e'''</font>, <font face=georgia>'''E'''</font>, <font face=georgia>'''D'''</font>, <font face=georgia>'''T'''</font>}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Operator
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
|
|-
| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
|-
| ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
|-
| (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
|-
| →
|-
| (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>])
|-
|
|-
|
|
|}
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''e'''</font>
|-
| <font face=georgia>'''E'''</font>
|-
| <font face=georgia>'''D'''</font>
|-
| <font face=georgia>'''T'''</font>
|}
|}
</font><br>
| valign="top" |
| colspan="2" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
| Radius Operator || <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>›
|-
| Secant Operator || <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E›
|-
| Chord Operator || <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D›
|-
| Tangent Functor || <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d›
|}
|}<br>
===Table 60. Propositional Transformation===
===Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes===
<pre>
<pre>
Table 60. Propositional Transformation
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
o-------------o-------------o-------------o-------------o
o--------------o----------------------o--------------------o----------------------o
| | Operator | Proposition | Transformation |
| | or | or | or |
| | | | |
| | Operand | Component | Mapping |
| 0 | 0 | 0 | 1 |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| | | | |
| Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
| | | | |
| | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
o-------------o-------------o-------------o-------------o
| Tacit | !e! : | !e!F_i : | !e!F : |
| | | ((u)(v)) | ((u, v)) |
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] |
| | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Trope | !h! : | !h!F_i : | !h!F : |
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| U |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Enlargement | E : | EF_i : | EF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | | |
| | | | |
| Difference | D : | DF_i : | DF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| o o o o |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| \ \ / / |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Differential | d : | dF_i : | dF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Remainder | r : | rF_i : | rF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Radius | $e$ = <!e!, !h!> : | | $e$F : |
| Operator | | | |
| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| | | | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| ////////o///////\ | |\\\\/ o \\\\\|
| | | | |
| //////////\///////\ | |\\\/ /\\ \\\\|
| Secant | $E$ = <!e!, E> : | | $E$F : |
| o///////o///o///////o | |\\o o\\\o o\\|
| Operator | | | |
| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| o///////o///o///////o | |\\o o\\\o o\\|
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| \///////o//////// | |\\\\\ o /\\\\|
| | | | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Chord | $D$ = <!e!, D> : | | $D$F : |
| Operator | | | |
| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| | | | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : |
| Functor | | | |
| | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| //////////////\ /\\\\\\\\\\\\\\ |
| | | | |
| | | B^n x D^n -> D | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
</pre>
</pre>
===Figure 62. Propositional Transformation (Short Form)===
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes'''
|- style="background:paleturquoise"
|
| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| align="center" | '''Operator<br>or<br>Operand'''
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| align="center" | '''Proposition<br>or<br>Component'''
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| align="center" | '''Transformation<br>or<br>Mapping'''
| ////////o///////\ | |\\\\/ o \\\\\|
|-
| //////////\///////\ | |\\\/ /\\ \\\\|
| Operand
| o///////o///o///////o | |\\o o\\\o o\\|
| valign="top" |
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| o///////o///o///////o | |\\o o\\\o o\\|
| ''F'' = ‹''F''<sub>1</sub>, ''F''<sub>2</sub>›
| \///////\////////// | |\\\\ \\/ /\\\|
|-
| \///////o//////// | |\\\\\ o /\\\\|
| ''F'' = ‹''f'', ''g''› : ''U'' → ''X''
| \////// \////// | |\\\\\\ /\\ /\\\\\|
|}
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| valign="top" |
| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| ''F''<sub>''i''</sub> : 〈''u'', ''v''〉 → '''B'''
|-
| ''F''<sub>''i''</sub> : '''B'''<sup>''n''</sup> → '''B'''
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
| ''F'' : [''u'', ''v''] → [''x'', ''y'']
|-
| ''F'' : '''B'''<sup>''n''</sup> → '''B'''<sup>''k''</sup>
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| o-----------o o-----------o |
| Tacit
| //////////////\ /\\\\\\\\\\\\\\ |
|-
| ////////////////o\\\\\\\\\\\\\\\\ |
| Extension
| /////////////////X\\\\\\\\\\\\\\\\\ |
|}
| /////////////////XXX\\\\\\\\\\\\\\\\\ |
|
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| <math>\epsilon</math> :
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
|-
| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → ''X''<sup> •</sup>)
| \///////////////\X/\\\\\\\\\\\\\\\/ |
|}
| \///////////////o\\\\\\\\\\\\\\\/ |
|
| \////////////// \\\\\\\\\\\\\\/ |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| o-----------o o-----------o |
| <math>\epsilon</math>''F''<sub>''i''</sub> :
| |
|-
| |
| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''B'''
</pre>
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\epsilon</math>''F'' :
<pre>
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Trope
|-
| Extension
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math> :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math>''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
| U | |\ | | |`U```````````````````````|
|}
| o---o o---o | | \ | | |``````o---o```o---o``````|
|
| /'''''\ /'''''\ | | \ | | |`````/ \`/ \`````|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math>''F'' :
| /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```|
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
| |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``|
|-
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
| \'''''''o'''''''/ | | \ | |````\ o /````|
|-
|
| o---o o---o | | | \ | |``````o---o```o---o``````|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| | | | \ * |`````````````````````````|
| Enlargement
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Difference
|-
| Operator
</pre>
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D :
<pre>
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D''F''<sub>''i''</sub> :
| 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F |
|-
| | | | | | | | = |
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| | | | | | | | |
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
| | | | | | | | | |
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Differential
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Remainder
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r''F'' :
</pre>
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
<br><font face="courier new">
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Radius
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
|-
|
|-
|
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''e'''</font>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Secant
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| width="8%" | E''G''<sub>''i''</sub>
|
| width="4%" | =
|-
| width="88%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
|
|-
|
|}
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''E'''</font>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Chord
|-
| Operator
|}
|}
|
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| width="8%" | D''G''<sub>''i''</sub>
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| width="4%" | +
| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
|-
|-
| width="8%" |
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|}
|
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| width="8%" | E''f''
|
| width="4%" | =
|-
| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))
|
|-
|-
| width="8%" | E''g''
|
| width="4%" | =
|}
| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''D'''</font>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Tangent
|-
| Functor
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''T'''</font>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|}
|}<br>
===Formula Display 17===
===Formula Display 12===
<pre>
<pre>
o-------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
| Df = ((u)(v)) + ((u + du)(v + dv)) |
| x = f(u, v) = ((u)(v)) |
| |
| |
| Dg = ((u, v)) + ((u + du, v + dv)) |
| y = g(u, v) = ((u, v)) |
| |
| |
o-------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
<br><font face="courier new">
<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| width="8%" | D''f''
|
| ''x''
| width="20%" | ((''u'')(''v''))
| =
| width="4%" | +
| ''f''‹''u'', ''v''›
| =
|-
| ((''u'')(''v''))
| width="8%" | D''g''
|
|-
| width="20%" | ((''u'', ''v''))
|
| width="4%" | +
| ''y''
| =
| ''g''‹''u'', ''v''›
| =
| ((''u'', ''v''))
|
|}
|}
|}
|}
</font><br>
</font><br>
===Table 66-i. Computation Summary for f‹u, v› = ((u)(v))===
===Formula Display 13===
<pre>
<pre>
o-----------------------------------------------------------o
o--------------------------------------------------------------------------------o
| |
| |
| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |
| !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 |
| |
o-----------------------------------------------------------o
| |
o--------------------------------------------------------------------------------o
</pre>
</pre>
<font face="courier new">
<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| <math>\epsilon</math>''f''
| ‹''x'', ''y''›
| =
| + || ''u''(''v'') || <math>\cdot</math> || 1
| ''F''‹''u'', ''v''›
| + || (''u'')''v'' || <math>\cdot</math> || 1
| =
| ‹((''u'')(''v'')), ((''u'', ''v''))›
|}
|}
| = || ''uv'' || <math>\cdot</math> || (d''u'' d''v'')
</font><br>
| + || ''u''(''v'') || <math>\cdot</math> || (d''u (d''v''))
<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|-
|
| D''f''
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|
| ‹''x'', ''y''›
| =
| ''F''‹''u'', ''v''›
|-
| =
| d''f''
| ‹((''u'')(''v'')), ((''u'', ''v''))›
| = || ''uv'' || <math>\cdot</math> || 0
|
| + || ''u''(''v'') || <math>\cdot</math> || d''u''
| + || (''u'')''v'' || <math>\cdot</math> || d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' d''v''
|}
|}
|}
|}
</font><br>
</font><br>
===Table 66-ii. Computation Summary for g‹u, v› = ((u, v))===
===Table 60. Propositional Transformation===
<pre>
<pre>
Table 66-ii. Computation Summary for g<u, v> = ((u, v))
Table 60. Propositional Transformation
o--------------------------------------------------------------------------------o
o-------------o-------------o-------------o-------------o
| |
| u | v | f | g |
| !e!g = uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 |
o-------------o-------------o-------------o-------------o
| |
| | | | |
| Eg = uv.((du, dv)) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v).((du, dv)) |
| 0 | 0 | 0 | 1 |
| |
| | | | |
| Dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
| 0 | 1 | 1 | 0 |
| |
| | | | |
| dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
| 1 | 0 | 1 | 0 |
| |
| | | | |
| rg = uv. 0 + u(v). 0 + (u)v. 0 + (u)(v). 0 |
| 1 | 1 | 1 | 1 |
| |
| | | | |
o--------------------------------------------------------------------------------o
o-------------o-------------o-------------o-------------o
| | | ((u)(v)) | ((u, v)) |
o-------------o-------------o-------------o-------------o
</pre>
</pre>
<font face="courier new">
<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ Table 66-ii. Computation Summary for g‹''u'', ''v''› = ((''u'', ''v''))
|+ '''Table 60. Propositional Transformation'''
|
|- style="background:paleturquoise"
| width="25%" | ''u''
| <math>\epsilon</math>''g''
| width="25%" | ''v''
| = || ''uv'' || <math>\cdot</math> || 1
| width="25%" | ''f''
| width="25%" | ''g''
| + || (''u'')''v'' || <math>\cdot</math> || 0
| + || (''u'')(''v'') || <math>\cdot</math> || 1
|-
|-
| E''g''
| width="25%" |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| 0
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'', d''v''))
|-
|-
| D''g''
| 0
|-
|-
| d''g''
| 1
|-
|-
| r''g''
| 1
|}
|}
| width="25%" |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 1
|-
| 0
|-
| 1
|}
|}
</font><br>
| width="25%" |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
===Table 67. Computation of an Analytic Series in Terms of Coordinates===
| 0
|-
| 1
|-
| 1
|-
| 1
|}
| width="25%" |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1
|-
| 0
|-
| 0
|-
| 1
|}
|-
| width="25%" |
| width="25%" |
| width="25%" | ((''u'')(''v''))
| width="25%" | ((''u'', ''v''))
|}
</font><br>
===Figure 61. Propositional Transformation===
<pre>
<pre>
o-----------------------------------------------------o
o--------o-------o-------o--------o-------o-------o-------o-------o
| U |
| |
o--------o-------o-------o--------o-------o-------o-------o-------o
| o-----------o o-----------o |
| | | | | | | | |
| / \ / \ |
| 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 |
| / o \ |
| | | | | | | | |
| / / \ \ |
| | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 |
| / / \ \ |
| | | | | | | | |
| o o o o |
| | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| | | | | | | | |
| \ \ / / |
| 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
| \ o / |
| | | | | | | | |
| \ / \ / |
| | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
| o-----------o o-----------o |
| | | | | | | | |
| |
| | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
| |
o-----------------------------------------------------o
| | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 |
/ \ / \
| | | | | | | | |
/ \ / \
o--------o-------o-------o--------o-------o-------o-------o-------o
/ \ / \
| | | | | | | | |
/ \ / \
| 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
/ \ / \
/ \ / \
| | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
/ \ / \
/ \ / \
| | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
/ \ / \
| | | | | | | | |
/ \ / \
/ \ / \
/ \ / \
o-------------------------o o-------------------------o
| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| 1 1 | 0 0 | 1 1 | 1 1 | 1 1 | 0 0 | 0 0 | 0 0 |
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| ////////o///////\ | |\\\\/ o \\\\\|
| //////////\///////\ | |\\\/ /\\ \\\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| \///////\////////// | |\\\\ \\/ /\\\|
| \///////o//////// | |\\\\\ o /\\\\|
| \////// \////// | |\\\\\\ /\\ /\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
o-------------------------o o-------------------------o
\ | | /
\ | | /
\ | | /
\ f | | g /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
o-------\----|---------------------------|----/-------o
| X \ | | / |
| \| |/ |
| o-----------o o-----------o |
| //////////////\ /\\\\\\\\\\\\\\ |
| ////////////////o\\\\\\\\\\\\\\\\ |
| /////////////////X\\\\\\\\\\\\\\\\\ |
| /////////////////XXX\\\\\\\\\\\\\\\\\ |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| \///////////////\X/\\\\\\\\\\\\\\\/ |
| \///////////////o\\\\\\\\\\\\\\\/ |
| \////////////// \\\\\\\\\\\\\\/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 61. Propositional Transformation
</pre>
</pre>
===Formula Display 18===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]</p>
<p><center><font size="+1">'''Figure 61. Propositional Transformation'''</font></center></p>
===Figure 62. Propositional Transformation (Short Form)===
<pre>
<pre>
o-------------------------------------------------------------------------o
o-------------------------o o-------------------------o
| |
| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| Df = uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) |
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| |
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| Dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv) |
| ////////o///////\ | |\\\\/ o \\\\\|
| |
| //////////\///////\ | |\\\/ /\\ \\\\|
o-------------------------------------------------------------------------o
| o///////o///o///////o | |\\o o\\\o o\\|
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| \///////\////////// | |\\\\ \\/ /\\\|
| \///////o//////// | |\\\\\ o /\\\\|
| \////// \////// | |\\\\\\ /\\ /\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
o-------------------------o o-------------------------o
\ / \ /
\ / \ /
\ / \ /
\ f / \ g /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o---------\-----/---------------------\-----/---------o
| X \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| //////////////\ /\\\\\\\\\\\\\\ |
| ////////////////o\\\\\\\\\\\\\\\\ |
| /////////////////X\\\\\\\\\\\\\\\\\ |
| /////////////////XXX\\\\\\\\\\\\\\\\\ |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| \///////////////\X/\\\\\\\\\\\\\\\/ |
| \///////////////o\\\\\\\\\\\\\\\/ |
| \////////////// \\\\\\\\\\\\\\/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 62. Propositional Transformation (Short Form)
</pre>
</pre>
<br><font face="courier new">
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]</p>
<p><center><font size="+1">'''Figure 62. Propositional Transformation (Short Form)'''</font></center></p>
===Figure 63. Transformation of Positions===
<pre>
o-----------------------------------------------------o
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' (d''v'')
|`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''
|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
|` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `|
|` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `|
|` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `|
|` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `|
|` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `|
|` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `|
|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
|` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `|
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
|-
|` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `|
|
|` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `|
|}
|` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `|
|}
|` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `|
|` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `|
|` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `|
|` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
|` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
o-----------\----|---------|---------|----------------o
" " \ | | | " "
" " \ | | | " "
" " \ | | | " "
" " \| | | " "
o-------------------------o \ | | o-------------------------o
| U | |\ | | |`U```````````````````````|
| ^ /`````````\|/`````````\ ^ | | ^ ```/ ^ \`/ ^ \``` ^ |
| o---o o---o | | \ | | |``````o---o```o---o``````|
| \ /```````````|```````````\ / | |``\``/ \ o / \``/``|
| /'''''\ /'''''\ | | \ | | |`````/ \`/ \`````|
| /'''''''o'''''''\ | | \ | | |````/ o \````|
| /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```|
| o``\````````o``@``o````````/``o | |``o \ o``@``o / o``|
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| |```\```````|`````|```````/```| | |``| \ |`````| / |``|
| |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``|
| |````@``````|`````|``````@````| | |``| @-------->`<--------@ |``|
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| \'''''''\'/'''''''/ | | \| | |```\ \`/ /```|
| o```````````o` ^ `o```````````o | |``o o`````o o``|
| \'''''''o'''''''/ | | \ | |````\ o /````|
| \```````````\`|`/```````````/ | |```\ \```/ /```|
| \'''''/ \'''''/ | | |\ | |`````\ /`\ /`````|
| \```` ^ ````\|/```` ^ ````/ | |````\ ^ \`/ ^ /````|
| o---o o---o | | | \ | |``````o---o```o---o``````|
| | | | \ * |`````````````````````````|
o-------------------------o | | \ / o-------------------------o
\ | | | \ / | /
\ ((u)(v)) | | | \/ | ((u, v)) /
| \ | / | |```````````````\```/```````````````|
\ | | | /\ | /
\ | | | / \ | /
\ | | | / \ | /
\ | | | / * | /
\ | | | / | | /
\ | | |/ | | /
\ | | / | | /
\ | | /| | | /
o-------\----|---|-------/-|---------|---|----/-------o
o----------\-------------/-----------------------\-------------/----------o
| X \ | | / | | | / |
| X \ / \ / |
| \| | / | | |/ |
| \ / \ / |
| o---|----/--o | o-------|---o |
| \ / \ / |
| /' ' | ' / ' '\|/` ` ` ` | ` `\ |
| o----------------o o----------------o |
| / ' ' | '/' ' ' | ` ` ` ` | ` ` \ |
| /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\ |
| / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \ |
| / / \ \ |
| @ o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o |
| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| / / \ \ |
| |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `| |
| / / \ \ |
| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o |
| o o o o |
| \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` / |
| \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/ |
| | | | | |
| \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` / |
| | f | | g | |
| \' ' ' ' ' ' '/ \` ` ` ` ` ` `/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 63. Transformation of Positions
Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
</pre>
</pre>
===Formula Display 19===
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]</p>
<p><center><font size="+1">'''Figure 63. Transformation of Positions'''</font></center></p>
===Table 64. Transformation of Positions===
<pre>
<pre>
o-------------------------------------------------------------------------------o
Table 64. Transformation of Positions
| |
o-----o----------o----------o-------o-------o--------o--------o-------------o
| df = uv. 0 + u(v). du + (u)v. dv + (u)(v).(du, dv) |
| u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] |
| |
o-----o----------o----------o-------o-------o--------o--------o-------------o
| dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv) |
| | | | | | | | ^ |
| 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | |
o-------------------------------------------------------------------------------o
| | | | | | | | |
| 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F |
| | | | | | | | = |
| 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> |
| | | | | | | | |
| 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ |
| | | | | | | | | |
o-----o----------o----------o-------o-------o--------o--------o-------------o
| | ((u)(v)) | ((u, v)) | u v | (u,v) | (u)(v) | 0 | U% = [u, v] |
o-----o----------o----------o-------o-------o--------o--------o-------------o
</pre>
</pre>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
|+ '''Table 64. Transformation of Positions'''
|- style="background:paleturquoise"
|
| ''u'' ''v''
| ''x''
| ''y''
| ''x'' ''y''
| ''x'' (''y'')
| (''x'') ''y''
| (''x'')(''y'')
| ''X''<sup> •</sup> = [''x'', ''y'' ]
|-
|-
| d''f''
| width="12%" |
| = || ''uv'' || <math>\cdot</math> || 0
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| + || ''u''(''v'') || <math>\cdot</math> || d''u''
| 0 0
|-
|-
|
| 0 1
|-
|-
| d''g''
| 1 0
|-
|-
|
| 1 1
|}
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 1
|-
| 1
|-
| 1
|}
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1
|-
| 0
|-
| 0
|-
| 1
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 0
|-
| 0
|-
| 1
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|\ / \ /| |\ / \ / / \ / / \ /|
| 0
| \ / \ / | | \ / \ / \ / \ / |
|-
| \ / \ / | | \ / O \ / O \ / O \ / |
| 1
| \ / \ / | | o /@ o @\ o /@ o |
|-
| \ / \ / | | |\ / \ / \ / \ / \ /| |
| 1
| \ / \ / | | | \ / \ / \ / | |
|-
| u \ / O \ / v | | u | \ / O \ / O \ / | v |
| 0
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1
|-
| 0
|-
| 0
|-
| 0
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 0
|-
| 0
|-
| 0
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| ↑
|-
| ''F''
|-
| ‹''f'', ''g'' ›
|-
| ↑
|}
|-
|
| ((''u'')(''v''))
| ((''u'', ''v''))
| ''u'' ''v''
| (''u'', ''v'')
| (''u'')(''v'')
| ( )
|\ / \ /| |\ / \ / \ / \ / \ / \ /|
| ''U''<sup> •</sup> = [''u'', ''v'' ]
| \ / \ / | | \ / \ / \ / \ / |
|}
| \ / \ / | | \ / O \ / O \ / O \ / |
<br>
| \ / \ / | | o /@ o @ o @\ o |
| \ / \ / | | |\ / / \ / \ / \ \ /| |
| \ / \ / | | | \ / \ / \ / | |
| x \ / O \ / y | | x | \ / O \ / O \ / | y |
===Table 65. Induced Transformation on Propositions===
===Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›===
<pre>
<pre>
o-----------------------o o-----------------------o o-----------------------o
Table 65. Induced Transformation on Propositions
| dU | | dU | | dU |
o------------o---------------------------------o------------o
| o--o o--o | | o--o o--o | | o--o o--o |
| X% | <--- F = <f , g> <--- | U% |
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |
o------------o----------o-----------o----------o------------o
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |
| | u = | 1 1 0 0 | = u | |
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| | v = | 1 0 1 0 | = v | |
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |
| f_i <x, y> o----------o-----------o----------o f_j <u, v> |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| | x = | 1 1 1 0 | = f<u,v> | |
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |
| | y = | 1 0 0 1 | = g<u,v> | |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
o------------o----------o-----------o----------o------------o
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |
| | | | | |
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| f_0 | () | 0 0 0 0 | () | f_0 |
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |
| | | | | |
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |
| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
| | | | | |
| f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |
o-----------------------o o-----------------------o o-----------------------o
| | | | | |
| f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 |
| | | | | |
| f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |
| | | | | |
| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |
| | | | | |
| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |
| | | | | |
| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |
| | | | | |
o------------o----------o-----------o----------o------------o
| | | | | |
| f_8 | x y | 1 0 0 0 | u v | f_8 |
| | | | | |
| f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |
| | | | | |
| f_10 | y | 1 0 1 0 | ((u, v)) | f_9 |
| | | | | |
| f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 |
| | | | | |
| f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 |
| | | | | |
| f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 |
| | | | | |
| f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 |
| | | | | |
| f_15 | (()) | 1 1 1 1 | (()) | f_15 |
| | | | | |
o------------o----------o-----------o----------o------------o
</pre>
o-----------------------o o-----------------------o o-----------------------o
<br><font face="courier new">
| dU | | dU | | dU |
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| o--o o--o | | o--o o--o | | o--o o--o |
|+ Table 65. Induced Transformation on Propositions
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |
|- style="background:paleturquoise"
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |
| ''X''<sup> •</sup>
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| colspan="3" |
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:80%"
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| ←
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |
| ''F'' = ‹''f'' , ''g''›
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| ←
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |
|}
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| ''U''<sup> •</sup>
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |
|- style="background:paleturquoise"
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |
| rowspan="2" | ''f''<sub>''i''</sub>‹''x'', ''y''›
| o--o o--o | | o--o o--o | | o--o o--o |
|
| | | | | |
{| align="right" style="background:paleturquoise; text-align:right"
o-----------------------o o-----------------------o o-----------------------o
| ''u'' =
= du' @ (u) v o-----------------------o dv' @ (u) v =
|-
= | dU' | =
| ''v'' =
= | o--o o--o | =
|}
= | /////\ /\\\\\ | =
|
= | ///////o\\\\\\\ | =
{| align="center" style="background:paleturquoise; text-align:center"
= | ////////X\\\\\\\\ | =
| 1 1 0 0
= | o///////XXX\\\\\\\o | =
|-
= | |/////oXXXXXo\\\\\| | =
| 1 0 1 0
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
|}
| |/////oXXXXXo\\\\\| |
|
| o//////\XXX/\\\\\\o |
{| align="left" style="background:paleturquoise; text-align:left"
| \//////\X/\\\\\\/ |
| = ''u''
| \//////o\\\\\\/ |
|-
| \///// \\\\\/ |
| = ''v''
| o--o o--o |
|}
| |
| rowspan="2" | ''f''<sub>''j''</sub>‹''u'', ''v''›
o-----------------------o
|- style="background:paleturquoise"
|
o-----------------------o o-----------------------o o-----------------------o
{| align="right" style="background:paleturquoise; text-align:right"
| dU | | dU | | dU |
| ''x'' =
| o--o o--o | | o--o o--o | | o--o o--o |
|-
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |
| ''y'' =
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |
|}
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
|
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |
{| align="center" style="background:paleturquoise; text-align:center"
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| 1 1 1 0
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |
|-
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| 1 0 0 1
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |
|}
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
|
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |
{| align="left" style="background:paleturquoise; text-align:left"
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |
| = ''f''‹''u'', ''v''›
| o--o o--o | | o--o o--o | | o--o o--o |
|-
| | | | | |
| = ''g''‹''u'', ''v''›
o-----------------------o o-----------------------o o-----------------------o
|}
= du' @ u (v) o-----------------------o dv' @ u (v) =
|-
= | dU' | =
|
= | o--o o--o | =
{| cellpadding="2" style="background:lightcyan"
= | /////\ /\\\\\ | =
| ''f''<sub>0</sub>
= | ///////o\\\\\\\ | =
|-
= | ////////X\\\\\\\\ | =
| ''f''<sub>1</sub>
= | o///////XXX\\\\\\\o | =
|-
= | |/////oXXXXXo\\\\\| | =
| ''f''<sub>2</sub>
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
|-
| |/////oXXXXXo\\\\\| |
| ''f''<sub>3</sub>
| o//////\XXX/\\\\\\o |
|-
| \//////\X/\\\\\\/ |
| ''f''<sub>4</sub>
| \//////o\\\\\\/ |
|-
| \///// \\\\\/ |
| ''f''<sub>5</sub>
| o--o o--o |
|-
| |
| ''f''<sub>6</sub>
o-----------------------o
|-
| ''f''<sub>7</sub>
o-----------------------o o-----------------------o o-----------------------o
|}
| dU | | dU | | dU |
|
| o--o o--o | | o--o o--o | | o--o o--o |
{| cellpadding="2" style="background:lightcyan"
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |
| ()
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |
|-
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| (''x'')(''y'')
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |
|-
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| (''x'') ''y''
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |
|-
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| (''x'')
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |
|-
| ''x'' (''y'')
|-
| (''y'')
|-
| (''x'', ''y'')
|-
| (''x'' ''y'')
|}
|
{| cellpadding="2" style="background:lightcyan"
| 0 0 0 0
|-
| 0 0 0 1
|-
| 0 0 1 0
|-
| 0 0 1 1
|-
| 0 1 0 0
|-
| 0 1 0 1
|-
| 0 1 1 0
|-
| 0 1 1 1
|}
|
{| cellpadding="2" style="background:lightcyan"
| ()
|-
| ()
|-
| (''u'')(''v'')
|-
| (''u'')(''v'')
|-
| (''u'', ''v'')
|-
| (''u'', ''v'')
|-
| (''u'' ''v'')
|-
| (''u'' ''v'')
|}
|
{| cellpadding="2" style="background:lightcyan"
| ''f''<sub>0</sub>
|-
| ''f''<sub>0</sub>
|-
| ''f''<sub>1</sub>
|-
| ''f''<sub>1</sub>
|-
| ''f''<sub>6</sub>
|-
| ''f''<sub>6</sub>
|-
| ''f''<sub>7</sub>
|-
| ''f''<sub>7</sub>
|}
|-
|
{| cellpadding="2" style="background:lightcyan"
| ''f''<sub>8</sub>
|-
| ''f''<sub>9</sub>
|-
| ''f''<sub>10</sub>
|-
| ''f''<sub>11</sub>
|-
| ''f''<sub>12</sub>
|-
| ''f''<sub>13</sub>
|-
| ''f''<sub>14</sub>
|-
| ''f''<sub>15</sub>
|}
|
{| cellpadding="2" style="background:lightcyan"
| ''x'' ''y''
|-
| ((''x'', ''y''))
|-
| ''y''
|-
| (''x'' (''y''))
|-
| ''x''
|-
| ((''x'') ''y'')
|-
| ((''x'')(''y''))
|-
| (())
|}
|
{| cellpadding="2" style="background:lightcyan"
| 1 0 0 0
|-
| 1 0 0 1
|-
| 1 0 1 0
|-
| 1 0 1 1
|-
| 1 1 0 0
|-
| 1 1 0 1
|-
| 1 1 1 0
|-
| 1 1 1 1
|}
|
{| cellpadding="2" style="background:lightcyan"
| ''u'' ''v''
|-
| ''u'' ''v''
|-
| ((''u'', ''v''))
|-
| ((''u'', ''v''))
|-
| ((''u'')(''v''))
|-
| ((''u'')(''v''))
|-
| (())
|-
| (())
|}
|
{| cellpadding="2" style="background:lightcyan"
| ''f''<sub>8</sub>
|-
| ''f''<sub>8</sub>
|-
| ''f''<sub>9</sub>
|-
| ''f''<sub>9</sub>
|-
| ''f''<sub>14</sub>
|-
| ''f''<sub>14</sub>
|-
| ''f''<sub>15</sub>
|-
| ''f''<sub>15</sub>
|}
|}
</font><br>
===Formula Display 14===
<pre>
o-------------------------------------------------o
| |
| EG_i = G_i <u + du, v + dv> |
| |
o-------------------------------------------------o
</pre>
<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | E''G''<sub>''i''</sub>
| width="4%" | =
| width="88%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
|}
|}
</font><br>
===Formula Display 15===
<pre>
o-------------------------------------------------o
| |
| DG_i = G_i <u, v> + EG_i <u, v, du, dv> |
| |
| = G_i <u, v> + G_i <u + du, v + dv> |
| |
o-------------------------------------------------o
</pre>
<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | D''G''<sub>''i''</sub>
| width="4%" | =
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
| width="4%" | +
| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
|-
| width="8%" |
| width="4%" | =
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
| width="4%" | +
| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
|}
|}
</font><br>
===Formula Display 16===
<pre>
o-------------------------------------------------o
| |
| Ef = ((u + du)(v + dv)) |
| |
| Eg = ((u + du, v + dv)) |
| |
o-------------------------------------------------o
</pre>
<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | E''f''
| width="4%" | =
| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))
|-
| width="8%" | E''g''
| width="4%" | =
| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))
|}
|}
</font><br>
===Formula Display 17===
<pre>
o-------------------------------------------------o
| |
| Df = ((u)(v)) + ((u + du)(v + dv)) |
| |
| Dg = ((u, v)) + ((u + du, v + dv)) |
| |
o-------------------------------------------------o
</pre>
<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | D''f''
| width="4%" | =
| width="20%" | ((''u'')(''v''))
| width="4%" | +
| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))
|-
| width="8%" | D''g''
| width="4%" | =
| width="20%" | ((''u'', ''v''))
| width="4%" | +
| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))
|}
|}
</font><br>
===Table 66-i. Computation Summary for f‹u, v› = ((u)(v))===
<pre>
Table 66-i. Computation Summary for f<u, v> = ((u)(v))
o--------------------------------------------------------------------------------o
| |
| !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 |
| |
| Ef = uv. (du dv) + u(v). (du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv)) |
| |
| Df = uv. du dv + u(v). du (dv) + (u)v. (du) dv + (u)(v).((du)(dv)) |
| |
| df = uv. 0 + u(v). du + (u)v. dv + (u)(v). (du, dv) |
| |
| rf = uv. du dv + u(v). du dv + (u)v. du dv + (u)(v). du dv |
| |
o--------------------------------------------------------------------------------o
</pre>
<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ Table 66-i. Computation Summary for ''f''‹''u'', ''v''› = ((''u'')(''v''))
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| <math>\epsilon</math>''f''
| = || ''uv'' || <math>\cdot</math> || 1
| + || ''u''(''v'') || <math>\cdot</math> || 1
| + || (''u'')''v'' || <math>\cdot</math> || 1
| + || (''u'')(''v'') || <math>\cdot</math> || 0
|-
| E''f''
| = || ''uv'' || <math>\cdot</math> || (d''u'' d''v'')
| + || ''u''(''v'') || <math>\cdot</math> || (d''u (d''v''))
| + || (''u'')''v'' || <math>\cdot</math> || ((d''u'') d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
|-
| D''f''
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' (d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
|-
| d''f''
| = || ''uv'' || <math>\cdot</math> || 0
| + || ''u''(''v'') || <math>\cdot</math> || d''u''
| + || (''u'')''v'' || <math>\cdot</math> || d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
| r''f''
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' d''v''
| + || (''u'')''v'' || <math>\cdot</math> || d''u'' d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
|}
|}
</font><br>
===Table 66-ii. Computation Summary for g‹u, v› = ((u, v))===
<pre>
Table 66-ii. Computation Summary for g<u, v> = ((u, v))
o--------------------------------------------------------------------------------o
| |
| !e!g = uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 |
| |
| Eg = uv.((du, dv)) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v).((du, dv)) |
| |
| Dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
| |
| dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
| |
| rg = uv. 0 + u(v). 0 + (u)v. 0 + (u)(v). 0 |
| |
o--------------------------------------------------------------------------------o
</pre>
<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ Table 66-ii. Computation Summary for g‹''u'', ''v''› = ((''u'', ''v''))
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| <math>\epsilon</math>''g''
| = || ''uv'' || <math>\cdot</math> || 1
| + || ''u''(''v'') || <math>\cdot</math> || 0
| + || (''u'')''v'' || <math>\cdot</math> || 0
| + || (''u'')(''v'') || <math>\cdot</math> || 1
|-
| E''g''
| = || ''uv'' || <math>\cdot</math> || ((d''u'', d''v''))
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'', d''v''))
|-
| D''g''
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
| d''g''
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
| r''g''
| = || ''uv'' || <math>\cdot</math> || 0
| + || ''u''(''v'') || <math>\cdot</math> || 0
| + || (''u'')''v'' || <math>\cdot</math> || 0
| + || (''u'')(''v'') || <math>\cdot</math> || 0
|}
|}
</font><br>
===Table 67. Computation of an Analytic Series in Terms of Coordinates===
<pre>
Table 67. Computation of an Analytic Series in Terms of Coordinates
o--------o-------o-------o--------o-------o-------o-------o-------o
| u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 0 1 | | 1 0 | 0 0 | 1 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 1 1 | 0 0 | 1 1 | 1 1 | 1 1 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 1 0 | | 1 0 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
</pre>
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ Table 67. Computation of an Analytic Series in Terms of Coordinates
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u''
| ''v''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| d''u''
| d''v''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u''<font face="courier new">’</font>
| ''v''<font face="courier new">’</font>
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|-
| 0 || 0
|-
| 1 || 1
|-
| 1 || 0
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|-
| 1 || 1
|-
| 0 || 0
|-
| 0 || 1
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 1
|-
| 1 || 0
|-
| 0 || 1
|-
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|}
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| <math>\epsilon</math>''f''
| <math>\epsilon</math>''g''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| E''f''
| E''g''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| D''f''
| D''g''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| d''f''
| d''g''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| d<sup>2</sup>''f''
| d<sup>2</sup>''g''
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|-
| 1 || 0
|-
| 1 || 0
|-
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 1 || 1
|-
| 1 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 1 || 1
|-
| 1 || 1
|-
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 0
|-
| 0 || 0
|-
| 1 || 0
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|-
| 0 || 1
|-
| 1 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 1 || 1
|-
| 0 || 1
|-
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 1 || 1
|-
| 0 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 0
|-
| 0 || 0
|-
| 1 || 0
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|-
| 1 || 1
|-
| 0 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 1
|-
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 0
|-
| 0 || 0
|-
| 1 || 0
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 1
|-
| 1 || 0
|-
| 1 || 0
|-
| 0 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 0 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 0 || 1
|-
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 0
|-
| 0 || 0
|-
| 1 || 0
|}
|}
|}
<br>
===Table 68. Computation of an Analytic Series in Symbolic Terms===
<pre>
Table 68. Computation of an Analytic Series in Symbolic Terms
o-----o-----o------------o----------o----------o----------o----------o----------o
| u v | f g | Df | Dg | df | dg | rf | rg |
o-----o-----o------------o----------o----------o----------o----------o----------o
| | | | | | | | |
| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () |
| | | | | | | | |
| 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () |
| | | | | | | | |
| 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () |
| | | | | | | | |
| 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () |
| | | | | | | | |
o-----o-----o------------o----------o----------o----------o----------o----------o
</pre>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 68. Computation of an Analytic Series in Symbolic Terms'''
|- style="background:paleturquoise"
| ''u'' ''v''
| ''f'' ''g''
| D''f''
| D''g''
| d''f''
| d''g''
| d<sup>2</sup>''f''
| d<sup>2</sup>''g''
|-
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 0
|-
| 0 1
|-
| 1 0
|-
| 1 1
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 1
|-
| 1 0
|-
| 1 0
|-
| 1 1
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| ((d''u'')(d''v''))
|-
| (d''u'') d''v''
|-
| d''u'' (d''v'')
|-
| d''u'' d''v''
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| (d''u'', d''v'')
|-
| d''v''
|-
| d''u''
|-
| ( )
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| d''u'' d''v''
|-
| d''u'' d''v''
|-
| d''u'' d''v''
|-
| d''u'' d''v''
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| ( )
|-
| ( )
|-
| ( )
|-
| ( )
|}
|}
<br>
===Formula Display 18===
<pre>
o-------------------------------------------------------------------------o
| |
| Df = uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) |
| |
| Dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv) |
| |
o-------------------------------------------------------------------------o
</pre>
<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|
|-
| D''f''
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' (d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
|-
|
|-
| D''g''
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
|
|}
|}
</font><br>
===Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›===
<pre>
o-----------------------------------o o-----------------------------------o
| U | |`U`````````````````````````````````|
| | |```````````````````````````````````|
| ^ | |```````````````````````````````````|
| | | |```````````````````````````````````|
| o-------o | o-------o | |```````o-------o```o-------o```````|
| ^ /`````````\|/`````````\ ^ | | ^ ```/ ^ \`/ ^ \``` ^ |
| \ /```````````|```````````\ / | |``\``/ \ o / \``/``|
| \/`````u`````/|\`````v`````\/ | |```\/ u \/`\/ v \/```|
| /\``````````/`|`\``````````/\ | |```/\ /\`/\ /\```|
| o``\````````o``@``o````````/``o | |``o \ o``@``o / o``|
| |```\```````|`````|```````/```| | |``| \ |`````| / |``|
| |````@``````|`````|``````@````| | |``| @-------->`<--------@ |``|
| |```````````|`````|```````````| | |``| |`````| |``|
| o```````````o` ^ `o```````````o | |``o o`````o o``|
| \```````````\`|`/```````````/ | |```\ \```/ /```|
| \```` ^ ````\|/```` ^ ````/ | |````\ ^ \`/ ^ /````|
| \`````\`````|`````/`````/ | |`````\ \ o / /`````|
| \`````\```/|\```/`````/ | |``````\ \ /`\ / /``````|
| o-----\-o | o-/-----o | |```````o-----\-o```o-/-----o```````|
| \ | / | |``````````````\`````/``````````````|
| \ | / | |```````````````\```/```````````````|
| \|/ | |````````````````\`/````````````````|
| @ | |`````````````````@`````````````````|
o-----------------------------------o o-----------------------------------o
\ / \ /
\ / \ /
\ ((u)(v)) / \ ((u, v)) /
\ / \ /
\ / \ /
o----------\-------------/-----------------------\-------------/----------o
| X \ / \ / |
| \ / \ / |
| \ / \ / |
| o----------------o o----------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | f | | g | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o----------------o o----------------o |
| |
| |
| |
o-------------------------------------------------------------------------o
Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]</p>
<p><center><font size="+1">'''Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›'''</font></center></p>
===Formula Display 19===
<pre>
o-------------------------------------------------------------------------------o
| |
| df = uv. 0 + u(v). du + (u)v. dv + (u)(v).(du, dv) |
| |
| dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv) |
| |
o-------------------------------------------------------------------------------o
</pre>
<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|
|-
| d''f''
| = || ''uv'' || <math>\cdot</math> || 0
| + || ''u''(''v'') || <math>\cdot</math> || d''u''
| + || (''u'')''v'' || <math>\cdot</math> || d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
|
|-
| d''g''
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
|
|}
|}
</font><br>
===Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›===
<pre>
o o
/ \ / \
/ \ / \
/ \ / O \
/ \ o /@\ o
/ \ / \ / \
/ \ / \ / \
/ O \ / O \ / O \
o /@\ o o /@\ o /@\ o
/ \ / \ / \ \ / \ \ / \
/ \ / \ / \ / \ / \
/ \ / \ / O \ / O \ / O \
/ \ / \ o /@ o /@\ o /@ o
/ \ / \ / \ \ / \ / \ \ / \
/ \ / \ / \ / \ / \ / \
/ O \ / O \ / O \ / O \ / O \ / O \
o /@ o /@ o o /@ o /@ o /@ o /@ o
|\ / \ /| |\ / \ / / \ / / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @\ o /@ o |
| \ / \ / | | |\ / \ / \ / \ / \ /| |
| \ / \ / | | | \ / \ / \ / | |
| u \ / O \ / v | | u | \ / O \ / O \ / | v |
o-------o @\ o-------o o---+---o @\ o @\ o---+---o
\ / | \ / \ / \ / \ / |
\ / | \ / \ / |
\ / | du \ / O \ / dv |
\ / o-------o @\ o-------o
\ / \ /
\ / \ /
\ / \ /
o o
U% $T$ $E$U%
o------------------>o
| |
| |
| |
| |
F | | $T$F
| |
| |
| |
v v
o------------------>o
X% $T$ $E$X%
o o
/ \ / \
/ \ / \
/ \ / O \
/ \ o /@\ o
/ \ / \ / \
/ \ / \ / \
/ O \ / O \ / O \
o /@\ o o /@\ o /@\ o
/ \ / \ / \ \ / \ / / \
/ \ / \ / \ / \ / \
/ \ / \ / O \ / O \ / O \
/ \ / \ o /@ o /@\ o @\ o
/ \ / \ / \ \ / \ / \ / \ / / \
/ \ / \ / \ / \ / \ / \
/ O \ / O \ / O \ / O \ / O \ / O \
o /@ o @\ o o /@ o /@ o @\ o @\ o
|\ / \ /| |\ / \ / \ / \ / \ / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @ o @\ o |
| \ / \ / | | |\ / / \ / \ / \ \ /| |
| \ / \ / | | | \ / \ / \ / | |
| x \ / O \ / y | | x | \ / O \ / O \ / | y |
o-------o @ o-------o o---+---o @ o @ o---+---o
\ / | \ / / \ \ / |
\ / | \ / \ / |
\ / | dx \ / O \ / dy |
\ / o-------o @ o-------o
\ / \ /
\ / \ /
\ / \ /
o o
Figure 70-a. Tangent Functor Diagram for F‹u, v› = <((u)(v)), ((u, v))>
</pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]</p>
<p><center><font size="+1">'''Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font></center></p>
===Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›===
<pre>
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u) v o-----------------------o dv' @ (u) v =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ u (v) o-----------------------o dv' @ u (v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
= du' @ u v o-----------------------o dv' @ u v =
= du' @ u v o-----------------------o dv' @ u v =
= | dU' | =
= | dU' | =
= | o--o o--o | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| o--o o--o |
| |
| |
o-----------------------o
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|
o-----------------------o o-----------------------o o-----------------------o
= u' o-----------------------o v' =
= | U' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
</pre>
[[Image:Tangent_Functor_Ferris_Wheel.gif|frame|<font size="3">'''Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font>]]
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
</pre>